Theorem onintexmid 4639

Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)

Hypothesis

Ref	Expression
onintexmid.onint	⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)

Assertion

Ref	Expression
onintexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem onintexmid

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prssi 3802	. . . . . 6 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → {𝑢, 𝑣} ⊆ On)
2		prmg 3765	. . . . . . 7 ⊢ (𝑢 ∈ On → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
3	2	adantr 276	. . . . . 6 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
4		zfpair2 4270	. . . . . . 7 ⊢ {𝑢, 𝑣} ∈ V
5		sseq1 3224	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → (𝑦 ⊆ On ↔ {𝑢, 𝑣} ⊆ On))
6		eleq2 2271	. . . . . . . . . 10 ⊢ (𝑦 = {𝑢, 𝑣} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑢, 𝑣}))
7	6	exbidv 1849	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → (∃𝑥 𝑥 ∈ 𝑦 ↔ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}))
8	5, 7	anbi12d 473	. . . . . . . 8 ⊢ (𝑦 = {𝑢, 𝑣} → ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) ↔ ({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣})))
9		inteq 3902	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → ∩ 𝑦 = ∩ {𝑢, 𝑣})
10		id 19	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → 𝑦 = {𝑢, 𝑣})
11	9, 10	eleq12d 2278	. . . . . . . 8 ⊢ (𝑦 = {𝑢, 𝑣} → (∩ 𝑦 ∈ 𝑦 ↔ ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣}))
12	8, 11	imbi12d 234	. . . . . . 7 ⊢ (𝑦 = {𝑢, 𝑣} → (((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦) ↔ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})))
13		onintexmid.onint	. . . . . . 7 ⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)
14	4, 12, 13	vtocl 2832	. . . . . 6 ⊢ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})
15	1, 3, 14	syl2anc 411	. . . . 5 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})
16		elpri 3666	. . . . 5 ⊢ (∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣} → (∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣))
17	15, 16	syl 14	. . . 4 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣))
18		incom 3373	. . . . . . 7 ⊢ (𝑣 ∩ 𝑢) = (𝑢 ∩ 𝑣)
19	18	eqeq1i 2215	. . . . . 6 ⊢ ((𝑣 ∩ 𝑢) = 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑢)
20		dfss1 3385	. . . . . 6 ⊢ (𝑢 ⊆ 𝑣 ↔ (𝑣 ∩ 𝑢) = 𝑢)
21		vex 2779	. . . . . . . 8 ⊢ 𝑢 ∈ V
22		vex 2779	. . . . . . . 8 ⊢ 𝑣 ∈ V
23	21, 22	intpr 3931	. . . . . . 7 ⊢ ∩ {𝑢, 𝑣} = (𝑢 ∩ 𝑣)
24	23	eqeq1i 2215	. . . . . 6 ⊢ (∩ {𝑢, 𝑣} = 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑢)
25	19, 20, 24	3bitr4ri 213	. . . . 5 ⊢ (∩ {𝑢, 𝑣} = 𝑢 ↔ 𝑢 ⊆ 𝑣)
26	23	eqeq1i 2215	. . . . . 6 ⊢ (∩ {𝑢, 𝑣} = 𝑣 ↔ (𝑢 ∩ 𝑣) = 𝑣)
27		dfss1 3385	. . . . . 6 ⊢ (𝑣 ⊆ 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑣)
28	26, 27	bitr4i 187	. . . . 5 ⊢ (∩ {𝑢, 𝑣} = 𝑣 ↔ 𝑣 ⊆ 𝑢)
29	25, 28	orbi12i 766	. . . 4 ⊢ ((∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣) ↔ (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
30	17, 29	sylib 122	. . 3 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
31	30	rgen2a 2562	. 2 ⊢ ∀𝑢 ∈ On ∀𝑣 ∈ On (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)
32	31	ordtri2or2exmid 4637	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710   = wceq 1373  ∃wex 1516   ∈ wcel 2178   ∩ cin 3173   ⊆ wss 3174  {cpr 3644  ∩ cint 3899  Oncon0 4428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-int 3900  df-tr 4159  df-iord 4431  df-on 4433  df-suc 4436

This theorem is referenced by: (None)